Exercises for Section 10.1.  Basic Properties of Conformal Mappings

Exercise 1.  State where the following mappings are conformal.  

1. (a)   [Graphics:Images/ConformalMappingModHome_gr_1.gif].  
Solution 1 (a).

 

1. (b)   [Graphics:Images/ConformalMappingModHome_gr_21.gif].  
Solution 1 (b).

 

1. (c)   [Graphics:Images/ConformalMappingModHome_gr_64.gif].  
Solution 1 (c).

 

1. (d)   [Graphics:Images/ConformalMappingModHome_gr_101.gif].  
Solution 1 (d).

 

1. (e)   [Graphics:Images/ConformalMappingModHome_gr_143.gif].  
Solution 1 (e).

 

1. (f)   [Graphics:Images/ConformalMappingModHome_gr_173.gif].  
Solution 1 (f).

 

 

Instructions For Exercises 2-5.   
Find the angle of rotation  [Graphics:Images/ConformalMappingModHome_gr_200.gif]  and the scale factor [Graphics:Images/ConformalMappingModHome_gr_201.gif]  of the mapping  [Graphics:Images/ConformalMappingModHome_gr_202.gif]  at the indicated points.  

Exercise 2.   [Graphics:Images/ConformalMappingModHome_gr_203.gif]  at the points  [Graphics:Images/ConformalMappingModHome_gr_204.gif].  
Solution 2.

 

Exercise 3.   [Graphics:Images/ConformalMappingModHome_gr_268.gif]  where  [Graphics:Images/ConformalMappingModHome_gr_269.gif]  at the points  [Graphics:Images/ConformalMappingModHome_gr_270.gif].  

Remark.  [Graphics:Images/ConformalMappingModHome_gr_271.gif].  
Solution 3.

 

Exercise 4.    [Graphics:Images/ConformalMappingModHome_gr_357.gif][Graphics:Images/ConformalMappingModHome_gr_358.gif],  where  [Graphics:Images/ConformalMappingModHome_gr_359.gif],  at the points  [Graphics:Images/ConformalMappingModHome_gr_360.gif].  

Remark.  [Graphics:Images/ConformalMappingModHome_gr_361.gif].  
Solution 4.

 

Exercise 5.  [Graphics:Images/ConformalMappingModHome_gr_443.gif]  at the points  [Graphics:Images/ConformalMappingModHome_gr_444.gif].  
Solution 5.

 

Exercise 6.  Consider the mapping  [Graphics:Images/ConformalMappingModHome_gr_522.gif].  

If  [Graphics:Images/ConformalMappingModHome_gr_523.gif],  show that the lines  [Graphics:Images/ConformalMappingModHome_gr_524.gif]  are mapped onto orthogonal parabolas.
Solution 6.

 

Exercise 7.  Consider the mapping  [Graphics:Images/ConformalMappingModHome_gr_588.gif],  where  [Graphics:Images/ConformalMappingModHome_gr_589.gif] denotes the principal branch of the square root function.  

If  [Graphics:Images/ConformalMappingModHome_gr_590.gif],  show that the lines  [Graphics:Images/ConformalMappingModHome_gr_591.gif]  are mapped onto orthogonal curves.
Solution 7.

 

Exercise 8.  Consider the mapping  [Graphics:Images/ConformalMappingModHome_gr_637.gif].  

Show that the lines  [Graphics:Images/ConformalMappingModHome_gr_638.gif]  are mapped onto orthogonal curves.
Solution 8.

 

Exercise 9.  Consider the mapping  [Graphics:Images/ConformalMappingModHome_gr_670.gif].  

Show that the line segment  [Graphics:Images/ConformalMappingModHome_gr_671.gif],  and the vertical line  [Graphics:Images/ConformalMappingModHome_gr_672.gif], where  [Graphics:Images/ConformalMappingModHome_gr_673.gif] are mapped onto orthogonal curves.  
Solution 9.

 

Exercise 10.  Consider the mapping  [Graphics:Images/ConformalMappingModHome_gr_722.gif],  where  [Graphics:Images/ConformalMappingModHome_gr_723.gif]  denotes the principal branch of the logarithm function.  

Show that the positive [Graphics:Images/ConformalMappingModHome_gr_724.gif]-axis and the vertical line  [Graphics:Images/ConformalMappingModHome_gr_725.gif]  are mapped onto orthogonal curves.  
Solution 10.

 

 

Exercise 11.  (Indirectly Conformal Mapping)  Let  [Graphics:Images/ConformalMappingModHome_gr_754.gif]   be analytic at  [Graphics:Images/ConformalMappingModHome_gr_755.gif]  and  [Graphics:Images/ConformalMappingModHome_gr_756.gif].  
Show that the function  [Graphics:Images/ConformalMappingModHome_gr_757.gif]  preserves the magnitude, but reverses the sense, of angles at  [Graphics:Images/ConformalMappingModHome_gr_758.gif].  

Solution 11.

 

 

Exercise 12.  If  [Graphics:Images/ConformalMappingModHome_gr_817.gif]  is a mapping, where  [Graphics:Images/ConformalMappingModHome_gr_818.gif]  is not analytic, then what behavior would one expect regarding the angles between curves?
Solution 12.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2008 John H. Mathews, Russell W. Howell